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Lie's reduction method and differential Galois theory in the complex analytic context
1. | Universidad Sergio Arboleda, Calle 74 no. 14-14, Bogotá, D.C., Colombia |
2. | Departamento de Matemática e Informática Aplicadas a la Ingeniería Civil, ETSI Caminos, Canales y Puertos, Universidad Politécnica de Madrid, Profesor Aranguren s/n (Ciudad Universitaria) - 28040 Madrid, Spain |
References:
[1] |
C. Athorne and T. Hartl, Solvable structures and hidden symmetries, J. Phys. A, 27 (1995), 3463-3474. |
[2] |
C. Athorne, Symmetries of linear ordinary differential equations, J. Phys. A, 30 (1997), 4639-4649.
doi: 10.1088/0305-4470/30/13/015. |
[3] |
C. Athorne, On the Lie symmetry algebra of general ordinary differential equation, J. Phys. A, 31 (1998), 6605-6614.
doi: 10.1088/0305-4470/31/31/008. |
[4] |
D. Blázquez-Sanz and J. J. Morales-Ruiz, Differential Galois theory of algebraic Lie-Vessiot systems, in "Differential Algebra, Complex Analysis and Orthogonal Polynomials,'' 1-58, Contemp. Math., 509, Amer. Math. Soc., Providence, RI, 2010. |
[5] |
D. Blázquez-Sanz and J. J. Morales-Ruiz, Local and global aspects of Lie superposition theorem, J. Lie Theory, 20 (2010), 483-517. |
[6] |
R. L. Bryant, "An introduction to Lie Groups and Symplectic Geometry,'' Lectures at the R.G.I. in Park City, Utah, 1991. |
[7] |
A. Buium, "Differential Function Fields and Moduli of Algebraic Varieties,'' Lecture Notes in Mathematics, 1226, Springer-Verlag, Berlin, 1986. |
[8] |
J. F. Cariñena, J. Grabowski and G. Marmo, "Lie-Scheffers Systems: A Geometric Approach,'' Napoli Series on Physics and Astrophysics, Bibliopolis, Naples, 2000. |
[9] |
J. F. Cariñena, J. Grabowski and G. Marmo, Superposition rules, Lie theorem, and partial differential equations, Rep. Math. Phys., 60 (2007), 237-258. |
[10] |
J. F. Cariñena, J. Grabowski and A. Ramos, Reduction of time-dependent systems admitting a superposition principle, Acta Appl. Math., 66 (2001), 67-87.
doi: 10.1023/A:1010743114995. |
[11] |
J. F. Cariñena, J. de Lucas and M. F. Rañada, Recent applications of the theory of Lie systems in Ermakov systems, SIGMA Symmetry Integrability Geom. Methods Appl., 4 (2008), Paper 031, 18 pp. |
[12] |
J. F. Cariñena and A. Ramos, A new geometric approach to Lie systems and physical applications. Symmetry and perturbation theory, Acta Appl. Math., 70 (2002), 43-69.
doi: 10.1023/A:1013913930134. |
[13] |
G. Casale, Feuilletages singuliers de codimension un, groupoïde de Galois et intégrales premières, (French) [Singular foliations of codimension one, Galois groupoids and first integrals], Ann. Inst. Fourier (Grenoble), 56 (2006), 735-779. |
[14] |
G. Casale, The Galois groupoid of Picard-Painlevé VI equation, "Algebraic, analytic and geometric aspects of complex differential equations and their deformations. Painlevé hierarchies,'' 15-20, RIMS Kôkyûroku Bessatsu, B2, Res. Inst. Math. Sci. (RIMS), Kyoto, 2007. |
[15] |
G. Casale, Le groupoïde de Galois de $P_1$ et son irréductibilité, Comment. Math. Helv., 83 (2008), 471-519.
doi: 10.4171/CMH/133. |
[16] |
G. Darboux, "Leçons sur la théorie générale des surfaces. I, II,'' (French) [Lessons on the general theory of surfaces. I, II] Reprint of the second (1914) edition (I) and the second (1915) edition (II), Les Grands Classiques Gauthier-Villars, Cours de Géométrie de la Faculté des Sciences, Éditions Jacques Gabay, Sceaux, 1993. |
[17] |
W. Fulton and J. Harris, "Representation Theory. A First Course,'' Graduate Texts in Mathematics, 129, Readings in Mathematics, Springer-Verlag, New York, 1991. |
[18] |
A. Guldberg, Sur les équations différentialles ordinaries qui possèdent un système fondamental d'intègrales, Compt. Rend. Acad. Sci. Paris, T CXVI, (1893), 964-965. |
[19] |
M. Havlíček, S. Pošta and P. Winternitz, Nonlinear superposition formulas based on imprimitive group action, J. Math. Phys., 40 (1999), 3104-3122.
doi: 10.1063/1.532749. |
[20] |
J. E. Humphreys, "Linear Algebraic Groups,'' Graduate Texts in Mathematics, No. 21, Springer-Verlag, New York-Heidelberg, 1975. |
[21] |
I. Kaplansky, "An Introduction to Differential Algebra,'' Actualités Sci. Ind., No. 1251, Hermann, Paris, 1957. |
[22] |
E. R. Kolchin, Galois theory of differential fields, Amer. J. Math., 75 (1953), 753-824.
doi: 10.2307/2372550. |
[23] |
E. R. Kolchin, "Differential Algebra and Algebraic Groups,'' Pure and Applied Mathematics, 54, Academic Press, New York-London, 1973. |
[24] |
A. G. Khovanskiĭ, Topological obstructions to the representability of functions by quadratures, J. Dynam. Control Systems, 1 (1995), 91-123.
doi: 10.1007/BF02254657. |
[25] |
S. Lie, Allgemeine Untersuchungen über Differentialgleichungen, die eine continuierliche endliche Gruppe gestatten, Math. Ann. Bd., 25 (1885), 71-151.
doi: 10.1007/BF01446421. |
[26] |
S. Lie, "Über Differentialgleichungen die Fundamentalintegrale besitzen,'' Lepziger Berichte, 1893. |
[27] |
S. Lie, "Vorlesungen über continuierliche Gruppen mit Geometrischen und anderen Anwendungen,'' (German) Bearbeitet und herausgegeben von Georg Scheffers, Nachdruck der Auflage des Jahres 1893, Chelsea Publishing Co., Bronx, N.Y., 1971. |
[28] |
S. Lie, Sur les équations différentielles ordinaries, qui possèddent des systemes fondamentaux d'integrales, Compt. Rend. Acad. Sci. Paris, T CXVI, (1893), 1233-1235. |
[29] |
S. Lie and G. Scheffers, "Vorlesungen über Differentialgleichungen mit bekannten Infinitesimalen Transformationen,'' Reprinted by Chelsea books, N.Y., 1967. |
[30] |
J. Liouville, Mémoire sur l'integration de une classe de équations différentielles du second ordre en quantités finies explicités, J. Math. Pures Appl., 4 (1839), 423-456. |
[31] |
B. Malgrange, Le groupoïde de Galois d'un feuilletage, (French) [The Galois groupoid of a foliation], in "Essays on Geometry and Related Topics," Vol. 1, 2, 465-501, Monogr. Enseign. Math., 38, Enseignement Math., Geneva, 2001. |
[32] |
B. Malgrange, On nonlinear differential Galois theory, Chinese Ann. Math. Ser. B, 23 (2002), 219-226.
doi: 10.1142/S0252959902000213. |
[33] |
B. Malgrange, "Pseudogroupes de Lie et Théorie de Galois Différentielle,'' (French) [Lie Pseudogroups and Differential Galois Theory], Prepublications I.H.E.S., 2010. |
[34] |
P. Malliavin, "Géométrie Différentielle Intrinsèque,'' (French) [Intrinsic Differential Geometry], Collection Enseignement des Sciences, No. 14, Hermann, Paris, 1972. |
[35] |
M. Maamache, Ermakov systems, exact solution, and geometrical angles and phases, Phys. Rev. A (3), 52 (1995), 936-940.
doi: 10.1103/PhysRevA.52.936. |
[36] |
J. J. Morales-Ruiz and J.-P. Ramis, Galoisian obstructions to integrability of Hamiltonian systems. I, II, Methods Appl. Anal., 8 (2001), 33-95, 97-111. |
[37] |
J. J. Morales-Ruiz and J.-P. Ramis, A note on the non-integrability of some Hamiltonian systems with a homogeneous potential, Methods Appl. Anal., 8 (2001), 113-120. |
[38] |
J. J. Morales-Ruiz, "Differential Galois Theory and Non-Integrability of Hamiltonian Systems,'' Progress in Mathematics, 179, Birkhäuser Verlag, Basel, 1999. |
[39] |
J. J. Morales-Ruiz, J.-P. Ramis and C. Simo, Integrability of Hamiltonian systems and differential Galois groups of higher variational equations, Ann. Sci. cole Norm. Sup. (4), 40 (2007), 845-884. |
[40] |
K. Nishioka, Differential algebraic function fields depending rationally on arbitrary constants, Nagoya Math. J., 113 (1989), 173-179. |
[41] |
K. Nishioka, General solutions depending algebraically on arbitrary constants, Nagoya Math. J., 113 (1989), 1-6. |
[42] |
K. Nishioka, Lie extensions, Proc. Japan Acad. Ser. A Math. Sci., 73 (1997), 82-85.
doi: 10.3792/pjaa.73.82. |
[43] |
K. Nomizu, "Lie Groups and Differential Geometry,'' The Mathematical Society of Japan, 1956. |
[44] |
J.-P. Ramis and J. Martinet, Théorie de Galois différentielle et resommation, (French) [Differential Galois theory and resummation], in "Computer Algebra and Differential Equations,'' 117-214, Comput. Math. Appl., Academic Press, London, 1990. |
[45] |
M. Rosenlicht, A remark on quotient spaces, An. Acad. Brasil. Ci., 35 (1963), 487-489. |
[46] |
C. Sancho de Salas, "Grupos Algebraicos y Teoría de Invariantes,'' (Spanish) [Algebraic Groups and Invariant Theory], Aportaciones Matemáticas: Textos [Mathematical Contributions: Texts], 16, Sociedad Matemática Mexicana, México, 2001. |
[47] |
J.-P. Serre, Géométrie algébrique et géométrie analytique (French), Ann. Inst. Fourier, Grenoble, 6 (1955-1956), 1-42. |
[48] |
J.-P. Serre, Espaces fibrés algebriques (French), Séminaire Claude Chevalley, 3 (1958), 1-37. |
[49] |
Y. Sibuya, "Linear Differential Equations in the Complex Domain: Problems of Analytic Continuation,'' Translated from the Japanese by the author, Translations of Mathematical Monographs, 82, American Mathematical Society, Providence, RI, 1990. |
[50] |
S. Shnider and P. Winternitz, Classification of systems of nonlinear ordinary differential equations with superposition principles, J. Math. Phys., 25 (1984), 3155-3165.
doi: 10.1063/1.526085. |
[51] |
M. Sorine and P. Winternitz, Superposition laws for solutions of differential matrix Riccati equations arising in control theory, IEEE Trans. Automat. Control, 30 (1985), 266-272.
doi: 10.1109/TAC.1985.1103934. |
[52] |
H. Umemura, On the irreducibility of the first differential equation of Painlevé, in "Algebraic Geometry and Commutative Algebra,'' Vol. II, 771-789, Kinokuniya, Tokyo, 1988. |
[53] |
H. Umemura, Galois theory of algebraic and differential equations, Nagoya Math. J., 144 (1996), 1-58. |
[54] |
H. Umemura, Differential Galois theory of infinite dimension, Nagoya Math. J., 144 (1996), 59-135. |
[55] |
H. Umemura, Sur l'équivalence des théories de Galois différentielles générales, (French) [On the equivalence of general differential Galois theories], C. R. Math. Acad. Sci. Paris, 346 (2008), 1155-1158. |
[56] |
M. van der Put and M. F. Singer, "Galois Theory of Linear Differential Equations,'' Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 328, Springer-Verlag, Berlin, 2003. |
[57] |
E. Vessiot, Sur l'intégration des equations différentielles linéaires (French), Ann. Sci. École Norm. Sup. (3), 9 (1892), 197-280. |
[58] |
E. Vessiot, Sur une classe d'équations différentielles, Ann. Sci. École Norm. Sup. (3), 10 (1893), 53-64. |
[59] |
E. Vessiot, Sur une classe systèmes d'équations différentielles ordinaires, Compt. Rend. Acad. Sci. Paris, T. CXVI, (1893), 1112-1114. |
[60] |
E. Vessiot, Sur les systèmes d'équations différentielles du premier ordre qui ont des systèmes fondamentaux d'intégrales, Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys., 8 (1894), H1-H33. |
[61] |
E. Vessiot, Sur la théorie de Galois et ses diverses généralisations (French), Ann. Sci. École Norm. Sup. (3), 21 (1904), 9-85. |
[62] |
E. Vessiot, Sur la réductibilité des systèmes automorphes dont le groupe d'automorphie est un groupe continu fini simplement transitif (French), Ann. École Norm. (3), 57 (1940), 1-60. |
show all references
References:
[1] |
C. Athorne and T. Hartl, Solvable structures and hidden symmetries, J. Phys. A, 27 (1995), 3463-3474. |
[2] |
C. Athorne, Symmetries of linear ordinary differential equations, J. Phys. A, 30 (1997), 4639-4649.
doi: 10.1088/0305-4470/30/13/015. |
[3] |
C. Athorne, On the Lie symmetry algebra of general ordinary differential equation, J. Phys. A, 31 (1998), 6605-6614.
doi: 10.1088/0305-4470/31/31/008. |
[4] |
D. Blázquez-Sanz and J. J. Morales-Ruiz, Differential Galois theory of algebraic Lie-Vessiot systems, in "Differential Algebra, Complex Analysis and Orthogonal Polynomials,'' 1-58, Contemp. Math., 509, Amer. Math. Soc., Providence, RI, 2010. |
[5] |
D. Blázquez-Sanz and J. J. Morales-Ruiz, Local and global aspects of Lie superposition theorem, J. Lie Theory, 20 (2010), 483-517. |
[6] |
R. L. Bryant, "An introduction to Lie Groups and Symplectic Geometry,'' Lectures at the R.G.I. in Park City, Utah, 1991. |
[7] |
A. Buium, "Differential Function Fields and Moduli of Algebraic Varieties,'' Lecture Notes in Mathematics, 1226, Springer-Verlag, Berlin, 1986. |
[8] |
J. F. Cariñena, J. Grabowski and G. Marmo, "Lie-Scheffers Systems: A Geometric Approach,'' Napoli Series on Physics and Astrophysics, Bibliopolis, Naples, 2000. |
[9] |
J. F. Cariñena, J. Grabowski and G. Marmo, Superposition rules, Lie theorem, and partial differential equations, Rep. Math. Phys., 60 (2007), 237-258. |
[10] |
J. F. Cariñena, J. Grabowski and A. Ramos, Reduction of time-dependent systems admitting a superposition principle, Acta Appl. Math., 66 (2001), 67-87.
doi: 10.1023/A:1010743114995. |
[11] |
J. F. Cariñena, J. de Lucas and M. F. Rañada, Recent applications of the theory of Lie systems in Ermakov systems, SIGMA Symmetry Integrability Geom. Methods Appl., 4 (2008), Paper 031, 18 pp. |
[12] |
J. F. Cariñena and A. Ramos, A new geometric approach to Lie systems and physical applications. Symmetry and perturbation theory, Acta Appl. Math., 70 (2002), 43-69.
doi: 10.1023/A:1013913930134. |
[13] |
G. Casale, Feuilletages singuliers de codimension un, groupoïde de Galois et intégrales premières, (French) [Singular foliations of codimension one, Galois groupoids and first integrals], Ann. Inst. Fourier (Grenoble), 56 (2006), 735-779. |
[14] |
G. Casale, The Galois groupoid of Picard-Painlevé VI equation, "Algebraic, analytic and geometric aspects of complex differential equations and their deformations. Painlevé hierarchies,'' 15-20, RIMS Kôkyûroku Bessatsu, B2, Res. Inst. Math. Sci. (RIMS), Kyoto, 2007. |
[15] |
G. Casale, Le groupoïde de Galois de $P_1$ et son irréductibilité, Comment. Math. Helv., 83 (2008), 471-519.
doi: 10.4171/CMH/133. |
[16] |
G. Darboux, "Leçons sur la théorie générale des surfaces. I, II,'' (French) [Lessons on the general theory of surfaces. I, II] Reprint of the second (1914) edition (I) and the second (1915) edition (II), Les Grands Classiques Gauthier-Villars, Cours de Géométrie de la Faculté des Sciences, Éditions Jacques Gabay, Sceaux, 1993. |
[17] |
W. Fulton and J. Harris, "Representation Theory. A First Course,'' Graduate Texts in Mathematics, 129, Readings in Mathematics, Springer-Verlag, New York, 1991. |
[18] |
A. Guldberg, Sur les équations différentialles ordinaries qui possèdent un système fondamental d'intègrales, Compt. Rend. Acad. Sci. Paris, T CXVI, (1893), 964-965. |
[19] |
M. Havlíček, S. Pošta and P. Winternitz, Nonlinear superposition formulas based on imprimitive group action, J. Math. Phys., 40 (1999), 3104-3122.
doi: 10.1063/1.532749. |
[20] |
J. E. Humphreys, "Linear Algebraic Groups,'' Graduate Texts in Mathematics, No. 21, Springer-Verlag, New York-Heidelberg, 1975. |
[21] |
I. Kaplansky, "An Introduction to Differential Algebra,'' Actualités Sci. Ind., No. 1251, Hermann, Paris, 1957. |
[22] |
E. R. Kolchin, Galois theory of differential fields, Amer. J. Math., 75 (1953), 753-824.
doi: 10.2307/2372550. |
[23] |
E. R. Kolchin, "Differential Algebra and Algebraic Groups,'' Pure and Applied Mathematics, 54, Academic Press, New York-London, 1973. |
[24] |
A. G. Khovanskiĭ, Topological obstructions to the representability of functions by quadratures, J. Dynam. Control Systems, 1 (1995), 91-123.
doi: 10.1007/BF02254657. |
[25] |
S. Lie, Allgemeine Untersuchungen über Differentialgleichungen, die eine continuierliche endliche Gruppe gestatten, Math. Ann. Bd., 25 (1885), 71-151.
doi: 10.1007/BF01446421. |
[26] |
S. Lie, "Über Differentialgleichungen die Fundamentalintegrale besitzen,'' Lepziger Berichte, 1893. |
[27] |
S. Lie, "Vorlesungen über continuierliche Gruppen mit Geometrischen und anderen Anwendungen,'' (German) Bearbeitet und herausgegeben von Georg Scheffers, Nachdruck der Auflage des Jahres 1893, Chelsea Publishing Co., Bronx, N.Y., 1971. |
[28] |
S. Lie, Sur les équations différentielles ordinaries, qui possèddent des systemes fondamentaux d'integrales, Compt. Rend. Acad. Sci. Paris, T CXVI, (1893), 1233-1235. |
[29] |
S. Lie and G. Scheffers, "Vorlesungen über Differentialgleichungen mit bekannten Infinitesimalen Transformationen,'' Reprinted by Chelsea books, N.Y., 1967. |
[30] |
J. Liouville, Mémoire sur l'integration de une classe de équations différentielles du second ordre en quantités finies explicités, J. Math. Pures Appl., 4 (1839), 423-456. |
[31] |
B. Malgrange, Le groupoïde de Galois d'un feuilletage, (French) [The Galois groupoid of a foliation], in "Essays on Geometry and Related Topics," Vol. 1, 2, 465-501, Monogr. Enseign. Math., 38, Enseignement Math., Geneva, 2001. |
[32] |
B. Malgrange, On nonlinear differential Galois theory, Chinese Ann. Math. Ser. B, 23 (2002), 219-226.
doi: 10.1142/S0252959902000213. |
[33] |
B. Malgrange, "Pseudogroupes de Lie et Théorie de Galois Différentielle,'' (French) [Lie Pseudogroups and Differential Galois Theory], Prepublications I.H.E.S., 2010. |
[34] |
P. Malliavin, "Géométrie Différentielle Intrinsèque,'' (French) [Intrinsic Differential Geometry], Collection Enseignement des Sciences, No. 14, Hermann, Paris, 1972. |
[35] |
M. Maamache, Ermakov systems, exact solution, and geometrical angles and phases, Phys. Rev. A (3), 52 (1995), 936-940.
doi: 10.1103/PhysRevA.52.936. |
[36] |
J. J. Morales-Ruiz and J.-P. Ramis, Galoisian obstructions to integrability of Hamiltonian systems. I, II, Methods Appl. Anal., 8 (2001), 33-95, 97-111. |
[37] |
J. J. Morales-Ruiz and J.-P. Ramis, A note on the non-integrability of some Hamiltonian systems with a homogeneous potential, Methods Appl. Anal., 8 (2001), 113-120. |
[38] |
J. J. Morales-Ruiz, "Differential Galois Theory and Non-Integrability of Hamiltonian Systems,'' Progress in Mathematics, 179, Birkhäuser Verlag, Basel, 1999. |
[39] |
J. J. Morales-Ruiz, J.-P. Ramis and C. Simo, Integrability of Hamiltonian systems and differential Galois groups of higher variational equations, Ann. Sci. cole Norm. Sup. (4), 40 (2007), 845-884. |
[40] |
K. Nishioka, Differential algebraic function fields depending rationally on arbitrary constants, Nagoya Math. J., 113 (1989), 173-179. |
[41] |
K. Nishioka, General solutions depending algebraically on arbitrary constants, Nagoya Math. J., 113 (1989), 1-6. |
[42] |
K. Nishioka, Lie extensions, Proc. Japan Acad. Ser. A Math. Sci., 73 (1997), 82-85.
doi: 10.3792/pjaa.73.82. |
[43] |
K. Nomizu, "Lie Groups and Differential Geometry,'' The Mathematical Society of Japan, 1956. |
[44] |
J.-P. Ramis and J. Martinet, Théorie de Galois différentielle et resommation, (French) [Differential Galois theory and resummation], in "Computer Algebra and Differential Equations,'' 117-214, Comput. Math. Appl., Academic Press, London, 1990. |
[45] |
M. Rosenlicht, A remark on quotient spaces, An. Acad. Brasil. Ci., 35 (1963), 487-489. |
[46] |
C. Sancho de Salas, "Grupos Algebraicos y Teoría de Invariantes,'' (Spanish) [Algebraic Groups and Invariant Theory], Aportaciones Matemáticas: Textos [Mathematical Contributions: Texts], 16, Sociedad Matemática Mexicana, México, 2001. |
[47] |
J.-P. Serre, Géométrie algébrique et géométrie analytique (French), Ann. Inst. Fourier, Grenoble, 6 (1955-1956), 1-42. |
[48] |
J.-P. Serre, Espaces fibrés algebriques (French), Séminaire Claude Chevalley, 3 (1958), 1-37. |
[49] |
Y. Sibuya, "Linear Differential Equations in the Complex Domain: Problems of Analytic Continuation,'' Translated from the Japanese by the author, Translations of Mathematical Monographs, 82, American Mathematical Society, Providence, RI, 1990. |
[50] |
S. Shnider and P. Winternitz, Classification of systems of nonlinear ordinary differential equations with superposition principles, J. Math. Phys., 25 (1984), 3155-3165.
doi: 10.1063/1.526085. |
[51] |
M. Sorine and P. Winternitz, Superposition laws for solutions of differential matrix Riccati equations arising in control theory, IEEE Trans. Automat. Control, 30 (1985), 266-272.
doi: 10.1109/TAC.1985.1103934. |
[52] |
H. Umemura, On the irreducibility of the first differential equation of Painlevé, in "Algebraic Geometry and Commutative Algebra,'' Vol. II, 771-789, Kinokuniya, Tokyo, 1988. |
[53] |
H. Umemura, Galois theory of algebraic and differential equations, Nagoya Math. J., 144 (1996), 1-58. |
[54] |
H. Umemura, Differential Galois theory of infinite dimension, Nagoya Math. J., 144 (1996), 59-135. |
[55] |
H. Umemura, Sur l'équivalence des théories de Galois différentielles générales, (French) [On the equivalence of general differential Galois theories], C. R. Math. Acad. Sci. Paris, 346 (2008), 1155-1158. |
[56] |
M. van der Put and M. F. Singer, "Galois Theory of Linear Differential Equations,'' Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 328, Springer-Verlag, Berlin, 2003. |
[57] |
E. Vessiot, Sur l'intégration des equations différentielles linéaires (French), Ann. Sci. École Norm. Sup. (3), 9 (1892), 197-280. |
[58] |
E. Vessiot, Sur une classe d'équations différentielles, Ann. Sci. École Norm. Sup. (3), 10 (1893), 53-64. |
[59] |
E. Vessiot, Sur une classe systèmes d'équations différentielles ordinaires, Compt. Rend. Acad. Sci. Paris, T. CXVI, (1893), 1112-1114. |
[60] |
E. Vessiot, Sur les systèmes d'équations différentielles du premier ordre qui ont des systèmes fondamentaux d'intégrales, Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys., 8 (1894), H1-H33. |
[61] |
E. Vessiot, Sur la théorie de Galois et ses diverses généralisations (French), Ann. Sci. École Norm. Sup. (3), 21 (1904), 9-85. |
[62] |
E. Vessiot, Sur la réductibilité des systèmes automorphes dont le groupe d'automorphie est un groupe continu fini simplement transitif (French), Ann. École Norm. (3), 57 (1940), 1-60. |
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